Relativistic X at Risk

PortfolioOptimisers.RelativisticValueatRisk Type

struct RelativisticValueatRisk{__T_settings, __T_slv, __T_alpha, __T_kappa, __T_w} <: RiskMeasure

Represents the Relativistic Value-at-Risk (RVaR) risk measure.

RelativisticValueatRisk is a coherent risk measure generalising EVaR via the Tsallis ($\kappa$-deformed) entropy. It is parametrised by a deformation parameter $\kappa \in (0,1)$ and reduces to EVaR in the limit $\kappa \to 0$. It is solved via a conic programme.

Mathematical Definition

Define the $\kappa$-logarithm ${\ell }_{\kappa }(u)=\frac{u^{\kappa }-u^{-\kappa }}{2\kappa }$. The RVaR is:

$${\mathrm{RVaR}}_{\alpha ,\kappa }(\mathit{x})=\underset{t,z\ge 0}{min}\{t+{\ell }_{\kappa }(\alpha T)z+\sum _{i=1}^T({\psi }_i+{\theta }_i)\}$$

subject to the power-cone constraints:

$$\begin{array}{rl}& (\frac{z(1+\kappa )}{2\kappa },\frac{{\psi }_i(1+\kappa )}{\kappa },{ϵ}_i)\in {\mathcal{K}}_{\mathrm{pow}}\left(\frac{1}{1+\kappa }\right)\mathrm{\forall }i,\\ & (\frac{{\omega }_i}{1-\kappa },\frac{{\theta }_i}{\kappa },-\frac{z}{2\kappa })\in {\mathcal{K}}_{\mathrm{pow}}(1-\kappa )\mathrm{\forall }i,\\ & {ϵ}_i+{\omega }_i\le x_i+t\mathrm{\forall }i,\end{array}$$

where ${\mathcal{K}}_{\mathrm{pow}}(p)=\{(a,b,c):a^p{b}^{1-p}\ge |c|,a\ge 0,b\ge 0\}$ is the power cone.

Fields

• settings: Risk measure configuration.

• slv: Solver or vector of solvers for the conic optimisation.

• alpha: Significance level for the lower tail.

• kappa: Deformation parameter $\kappa \in (0,1)$.

• w: Optional observation weights.

Constructors

RelativisticValueatRisk(;
    settings::RiskMeasureSettings = RiskMeasureSettings(),
    slv::Option{<:Slv_VecSlv} = nothing,
    alpha::Number = 0.05,
    kappa::Number = 0.3,
    w::Option{<:ObsWeights} = nothing
) -> RelativisticValueatRisk

Keywords correspond to the struct's fields.

Validation

• 0 < alpha < 1.

• 0 < kappa < 1.

• If slv is a VecSlv: !isempty(slv).

• If w is not nothing: !isempty(w).

Functor

(r::RelativisticValueatRisk)(x::VecNum)

Computes the RVaR of a portfolio returns vector x.

Arguments

• x::VecNum: Portfolio returns vector.

Examples

julia> RelativisticValueatRisk()
RelativisticValueatRisk
  settings ┼ RiskMeasureSettings
           │   scale ┼ Float64: 1.0
           │      ub ┼ nothing
           │     rke ┴ Bool: true
       slv ┼ nothing
     alpha ┼ Float64: 0.05
     kappa ┼ Float64: 0.3
         w ┴ nothing

Related

• RiskMeasure

• RiskMeasureSettings

• EntropicValueatRisk

• RelativisticValueatRiskRange

• RelativisticDrawdownatRisk

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PortfolioOptimisers.RelativisticValueatRiskRange Type

struct RelativisticValueatRiskRange{__T_settings, __T_slv, __T_alpha, __T_kappa_a, __T_beta, __T_kappa_b, __T_w} <: RiskMeasure

Represents the Relativistic Value-at-Risk Range (RVaR Range) risk measure.

RelativisticValueatRiskRange computes the sum of the lower-tail RVaR (at level alpha with deformation kappa_a) and the upper-tail RVaR (at level beta with deformation kappa_b).

Mathematical Definition

$${\mathrm{RVaRRange}}_{\alpha ,{\kappa }_a,\beta ,{\kappa }_b}(\mathit{x})={\mathrm{RVaR}}_{\alpha ,{\kappa }_a}(\mathit{x})+{\mathrm{RVaR}}_{\beta ,{\kappa }_b}(-\mathit{x}).$$

Fields

• settings: Risk measure configuration.

• slv: Solver or vector of solvers for the conic optimisation.

• alpha: Significance level for the lower tail.

• kappa_a: Deformation parameter for the lower-tail RVaR.

• beta: Significance level for the upper tail.

• kappa_b: Deformation parameter for the upper-tail RVaR.

• w: Optional observation weights.

Constructors

RelativisticValueatRiskRange(;
    settings::RiskMeasureSettings = RiskMeasureSettings(),
    slv::Option{<:Slv_VecSlv} = nothing,
    alpha::Number = 0.05,
    kappa_a::Number = 0.3,
    beta::Number = 0.05,
    kappa_b::Number = 0.3,
    w::Option{<:ObsWeights} = nothing
) -> RelativisticValueatRiskRange

Keywords correspond to the struct's fields.

Validation

• 0 < alpha < 1, 0 < kappa_a < 1.

• 0 < beta < 1, 0 < kappa_b < 1.

• If slv is a VecSlv: !isempty(slv).

• If w is not nothing: !isempty(w).

Functor

(r::RelativisticValueatRiskRange)(x::VecNum)

Computes the RVaR Range of a portfolio returns vector x.

Arguments

• x::VecNum: Portfolio returns vector.

Examples

julia> RelativisticValueatRiskRange()
RelativisticValueatRiskRange
  settings ┼ RiskMeasureSettings
           │   scale ┼ Float64: 1.0
           │      ub ┼ nothing
           │     rke ┴ Bool: true
       slv ┼ nothing
     alpha ┼ Float64: 0.05
   kappa_a ┼ Float64: 0.3
      beta ┼ Float64: 0.05
   kappa_b ┼ Float64: 0.3
         w ┴ nothing

Related

• RiskMeasure

• RiskMeasureSettings

• RelativisticValueatRisk

• EntropicValueatRiskRange

source

PortfolioOptimisers.RelativisticDrawdownatRisk Type

struct RelativisticDrawdownatRisk{__T_settings, __T_slv, __T_alpha, __T_kappa, __T_w} <: RiskMeasure

Represents the Relativistic Drawdown-at-Risk (RDDaR) risk measure.

RelativisticDrawdownatRisk applies the Relativistic Value-at-Risk framework to the absolute drawdown series of portfolio returns.

Mathematical Definition

Define the absolute drawdown series:

$$c_t=\sum _{s=1}^t{x}_s,d_t=c_t-\underset{0\le s\le t}{max}{c}_s\le 0.$$

The Relativistic Drawdown-at-Risk is the RVaR of the drawdown series:

$${\mathrm{RDDaR}}_{\alpha ,\kappa }(\mathit{x})={\mathrm{RVaR}}_{\alpha ,\kappa }(\mathit{d}(\mathit{x})).$$

Fields

• settings: Risk measure configuration.

• slv: Solver or vector of solvers for the conic optimisation.

• alpha: Significance level for the lower tail.

• kappa: Deformation parameter $\kappa \in (0,1)$.

• w: Optional observation weights.

Constructors

RelativisticDrawdownatRisk(;
    settings::RiskMeasureSettings = RiskMeasureSettings(),
    slv::Option{<:Slv_VecSlv} = nothing,
    alpha::Number = 0.05,
    kappa::Number = 0.3,
    w::Option{<:ObsWeights} = nothing
) -> RelativisticDrawdownatRisk

Keywords correspond to the struct's fields.

Validation

• 0 < alpha < 1.

• 0 < kappa < 1.

• If slv is a VecSlv: !isempty(slv).

• If w is not nothing: !isempty(w).

Functor

(r::RelativisticDrawdownatRisk)(x::VecNum)

Computes the Relativistic Drawdown-at-Risk of a portfolio returns vector x.

Arguments

• x::VecNum: Portfolio returns vector.

Examples

julia> RelativisticDrawdownatRisk()
RelativisticDrawdownatRisk
  settings ┼ RiskMeasureSettings
           │   scale ┼ Float64: 1.0
           │      ub ┼ nothing
           │     rke ┴ Bool: true
       slv ┼ nothing
     alpha ┼ Float64: 0.05
     kappa ┼ Float64: 0.3
         w ┴ nothing

Related

• RiskMeasure

• RiskMeasureSettings

• RelativisticValueatRisk

• EntropicDrawdownatRisk

• RelativeRelativisticDrawdownatRisk

source

PortfolioOptimisers.RelativeRelativisticDrawdownatRisk Type

struct RelativeRelativisticDrawdownatRisk{__T_settings, __T_slv, __T_alpha, __T_kappa, __T_w} <: HierarchicalRiskMeasure

Represents the Relative Relativistic Drawdown-at-Risk (Relative RDDaR) risk measure for hierarchical optimisation.

RelativeRelativisticDrawdownatRisk applies the Relativistic Value-at-Risk framework to the relative (compounded) drawdown series of portfolio returns.

Mathematical Definition

Define the compounded wealth process and relative drawdown series:

$$C_t=\prod _{s=1}^t(1+x_s),r{d}_t=\frac{C_t}{\underset{0\le s\le t}{max}{C}_s}-1\le 0.$$

The Relative Relativistic Drawdown-at-Risk is the RVaR of the relative drawdown series:

$${\mathrm{RRDDaR}}_{\alpha ,\kappa }(\mathit{x})={\mathrm{RVaR}}_{\alpha ,\kappa }(\mathit{r}\mathit{d}(\mathit{x})).$$

Fields

• settings: Hierarchical risk measure configuration.

• slv: Solver or vector of solvers for the conic optimisation.

• alpha: Significance level for the lower tail.

• kappa: Deformation parameter $\kappa \in (0,1)$.

• w: Optional observation weights.

Constructors

RelativeRelativisticDrawdownatRisk(;
    settings::HierarchicalRiskMeasureSettings = HierarchicalRiskMeasureSettings(),
    slv::Option{<:Slv_VecSlv} = nothing,
    alpha::Number = 0.05,
    kappa::Number = 0.3,
    w::Option{<:ObsWeights} = nothing
) -> RelativeRelativisticDrawdownatRisk

Keywords correspond to the struct's fields.

Validation

• 0 < alpha < 1.

• 0 < kappa < 1.

• If slv is a VecSlv: !isempty(slv).

• If w is not nothing: !isempty(w).

Functor

(r::RelativeRelativisticDrawdownatRisk)(x::VecNum)

Computes the Relative Relativistic Drawdown-at-Risk of a portfolio returns vector x.

Arguments

• x::VecNum: Portfolio returns vector.

Examples

julia> RelativeRelativisticDrawdownatRisk()
RelativeRelativisticDrawdownatRisk
  settings ┼ HierarchicalRiskMeasureSettings
           │   scale ┴ Float64: 1.0
       slv ┼ nothing
     alpha ┼ Float64: 0.05
     kappa ┼ Float64: 0.3
         w ┴ nothing

Related

• HierarchicalRiskMeasure

• HierarchicalRiskMeasureSettings

• RelativisticDrawdownatRisk

• RelativeEntropicDrawdownatRisk

source